Problem 17 - Algebra and Trigonometry

Question

Evaluate each expression. (a) \(100^{-1.5}\) (b) \(4^{2 / 3} \cdot 6^{2 / 3} \cdot 9^{2 / 3}\) (c) \(0.001^{-2 / 3}\)

Step-by-Step Solution

Answer

(a) 0.001, (b) 36, (c) 100.

Step 1: Evaluating Part (a)

To evaluate \( 100^{-1.5} \), we can rewrite it using the property of exponents \( a^{-b} = \frac{1}{a^b} \). Hence, \( 100^{-1.5} = \frac{1}{100^{1.5}} \). Now compute \( 100^{1.5} \) by rewriting it as \( 100^{1+0.5} = 100^1 \times 100^{0.5} \).\( 100^1 \) is simply 100 and \( 100^{0.5} = \sqrt{100} = 10 \). Therefore, \( 100^{1.5} = 100 \times 10 = 1000 \). Thus, \( 100^{-1.5} = \frac{1}{1000} = 0.001 \).

Step 2: Evaluating Part (b)

For \( 4^{2/3} \cdot 6^{2/3} \cdot 9^{2/3} \), use the property \( a^{m/n} = \sqrt[n]{a^m} \), which means \((abc)^{m} = a^m \cdot b^m \cdot c^m\) if \(m\) is the same for all factors. Here \((abc)^{2/3}\) where \(abc = 4 \cdot 6 \cdot 9 = 216\), computes to \(216^{2/3}\). Calculating \(216^{2/3}\) first involves finding the cube root of 216 which is \(\sqrt[3]{216} = 6\) and squaring it gives \((6)^2 = 36\). The expression evaluates to 36.

Step 3: Evaluating Part (c)

For \(0.001^{-2/3}\), rewrite it as \((0.001)^{-2/3} = \frac{1}{0.001^{2/3}}\). First, find the cube root of 0.001, which is \(\sqrt[3]{0.001} = 0.1 \), and raise it to the power of 2: \((0.1)^2 = 0.01\). Hence, \(0.001^{2/3} = 0.01\), leading to \(0.001^{-2/3} = \frac{1}{0.01} = 100 \).

Key Concepts

Negative exponents

Negative exponents can sometimes seem a bit challenging, but they are just a variation of our regular exponent rules. When you see a negative exponent, such as in the expression \(100^{-1.5}\), it signifies the reciprocal of the base raised to the positive of that exponent. To put it simply:

• \(a^{-b} = \frac{1}{a^b}\)

For example, \(100^{-1.5}\) becomes \(\frac{1}{100^{1.5}}\). You handle the exponent as usual, by first converting it to a positive form and calculating as you normally would.

In exercises where you encounter this, remember that the negative exponent flips the base to the denominator, and you proceed by calculating the base raised to the positive version of that exponent.

Fractional exponents

Fractional exponents might seem daunting at first, but they can be easily understood as roots and powers. When you come across an expression like \(a^{m/n}\), it indicates a root and a power:

• The numerator \(m\) represents the power.
• The denominator \(n\) signifies the root.

This means \(a^{m/n} = \sqrt[n]{a^m}\). For instance, in the expression \((4^{2/3})\), it translates to finding the cube root of 4 squared. A handy way to remember this is:

• First, find the root identified by the denominator.
• Then, raise the result to the power signified by the numerator.

Using our solution as an example, we combined numbers before applying the exponent: \((abc)^{2/3}\), simplifying our work and leading to an easier computation, \(6^2 = 36\).

Properties of exponents

The properties of exponents are rules that assist in simplifying and solving expressions involving powers. Here are a few key properties:

• \(a^m \cdot a^n = a^{m+n}\): Multiply same bases adds the exponents.
• \((a^m)^n = a^{m \cdot n}\): Raising an exponent to another exponent multiplies them.
• \(a^{m/n} = \sqrt[n]{a^m}\): A fractional exponent means roots and powers combined.

When evaluating something like \(0.001^{-2/3}\), you would utilize these properties by first changing the negative exponent into a reciprocal as discussed in section one. Then, apply the fractional exponent as explained earlier.

Mastering these properties makes working with exponents straightforward, allowing you to transform complex expressions into simpler, solvable forms.